Properties

Label 6003.2501
Modulus $6003$
Conductor $2001$
Order $154$
Real no
Primitive no
Minimal no
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6003, base_ring=CyclotomicField(154))
 
M = H._module
 
chi = DirichletCharacter(H, M([77,49,66]))
 
pari: [g,chi] = znchar(Mod(2501,6003))
 

Basic properties

Modulus: \(6003\)
Conductor: \(2001\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(154\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{2001}(500,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 6003.cx

\(\chi_{6003}(53,\cdot)\) \(\chi_{6003}(107,\cdot)\) \(\chi_{6003}(152,\cdot)\) \(\chi_{6003}(314,\cdot)\) \(\chi_{6003}(431,\cdot)\) \(\chi_{6003}(458,\cdot)\) \(\chi_{6003}(674,\cdot)\) \(\chi_{6003}(935,\cdot)\) \(\chi_{6003}(953,\cdot)\) \(\chi_{6003}(980,\cdot)\) \(\chi_{6003}(1466,\cdot)\) \(\chi_{6003}(1502,\cdot)\) \(\chi_{6003}(1673,\cdot)\) \(\chi_{6003}(1736,\cdot)\) \(\chi_{6003}(1763,\cdot)\) \(\chi_{6003}(1880,\cdot)\) \(\chi_{6003}(1988,\cdot)\) \(\chi_{6003}(1997,\cdot)\) \(\chi_{6003}(2195,\cdot)\) \(\chi_{6003}(2402,\cdot)\) \(\chi_{6003}(2501,\cdot)\) \(\chi_{6003}(2771,\cdot)\) \(\chi_{6003}(2780,\cdot)\) \(\chi_{6003}(2978,\cdot)\) \(\chi_{6003}(3023,\cdot)\) \(\chi_{6003}(3032,\cdot)\) \(\chi_{6003}(3041,\cdot)\) \(\chi_{6003}(3185,\cdot)\) \(\chi_{6003}(3239,\cdot)\) \(\chi_{6003}(3329,\cdot)\) ...

sage: chi.galois_orbit()
 
order = charorder(g,chi)
 
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: $\Q(\zeta_{77})$
Fixed field: Number field defined by a degree 154 polynomial (not computed)

Values on generators

\((668,3133,4555)\) → \((-1,e\left(\frac{7}{22}\right),e\left(\frac{3}{7}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 6003 }(2501, a) \) \(1\)\(1\)\(e\left(\frac{87}{154}\right)\)\(e\left(\frac{10}{77}\right)\)\(e\left(\frac{19}{77}\right)\)\(e\left(\frac{29}{154}\right)\)\(e\left(\frac{107}{154}\right)\)\(e\left(\frac{125}{154}\right)\)\(e\left(\frac{6}{77}\right)\)\(e\left(\frac{13}{77}\right)\)\(e\left(\frac{58}{77}\right)\)\(e\left(\frac{20}{77}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6003 }(2501,a) \;\) at \(\;a = \) e.g. 2